In the studio, the host and producer(s) can see if the recordings are completely uploaded by seeing the word "done" next to the user's name in the recordings window on the right-hand side of their studio view (where the 33/38 MB is displayed in the picture below):
Any of the recorded participants can check their own upload status by hovering over the upload icon in the top left of the studio.
An incomplete upload is showing an upward pointing arrow icon:
A complete upload will display a checkmark:
Even if an upload's progress is at 100%, it's still uploading! Please wait for the checkmark to show up to close your own Riverside.fm tab.